Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

Abstract

We introduce a generalization of the set of completely positive matrices that we call "pairwise completely positive" (PCP) matrices. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. We explore basic properties of these matrix pairs and develop several testable necessary and sufficient conditions that help determine whether or not a pair is PCP. We then establish a connection with quantum entanglement by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of our results, we show that a wide family of states that have absolutely positive partial transpose are in fact separable.

arXiv comment26 pages, comments welcome

References in corpus (7)

• Separability Criterion for Density Matrices
• Symmetric Informationally Complete Quantum Measurements
• Maximally entangled mixed states of two qubits
• Informationally complete measurements and groups representation
• A separability criterion for density operators
• Separability from Spectrum for Qubit-Qudit States
• PPT from spectra

Cited by in corpus (10)

• Diagonal unitary and orthogonal symmetries in quantum theory
• Detecting positive quantum capacities of quantum channels
• The PPT${}^2$ conjecture holds for all Choi-type maps
• A graphical calculus for integration over random diagonal unitary matrices
• Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators
• Can entanglement hide behind triangle-free graphs?
• Random covariant quantum channels
• Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)
• Entanglement in the Dicke subspace
• Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products